We present strongly convergent explicit and semi-implicit adaptive numerical schemes for systems of semi-linear stochastic differential equations (SDEs) where both the drift and diffusion are not globally Lipschitz continuous. Numerical instability may arise either from the stiffness of the linear operator or from the perturbation of the nonlinear drift under discretization, or both. Typical applications arise from the space discretization of an SPDE, stochastic volatility models in finance, or certain ecological models. Under conditions that include montonicity, we prove that a timestepping strategy which adapts the stepsize based on the drift alone is sufficient to control growth and to obtain strong convergence with polynomial order. The order of strong convergence of our scheme is (1 − ε)/2, for ε ∈ (0,1), where ε becomes arbitrarily small as the number of finite moments available for solutions of the SDE increases. Numerically, we compare the adaptive semi-implicit method to a fully drift-implicit method and to three other explicit methods. Our numerical results show that overall the adaptive semi-implicit method is robust, efficient, and well suited as a general purpose solver.

我们为漂移和扩散都不是全局 Lipschitz 连续的半线性随机微分方程 (SDE) 系统提出了强收敛显式和半隐式自适应数值方案。数值不稳定性可能来自线性算子的刚度或离散化下非线性漂移的扰动，或两者兼而有之。典型应用来自 SPDE 的空间离散化、金融中的随机波动模型或某些生态模型。在包括单调性的条件下，我们证明了仅基于漂移来调整步长的时间步长策略足以控制增长并获得多项式阶的强收敛。我们方案的强收敛阶数是 (1 − ε )/2，对于ε ∈ (0,1)，其中随着可用于 SDE 解的有限矩数量的增加，ε变得任意小。在数值上，我们将自适应半隐式方法与完全漂移隐式方法和其他三种显式方法进行了比较。我们的数值结果表明，总体而言，自适应半隐式方法是稳健、高效的，并且非常适合作为通用求解器。